Fitting a Curve to an Uncurling Object

Hello all, I am currently conducting video analysis on an uncurling object and would like to fit a curve to the object along its path as it straightens to later calculate curvature along the object's length over time. I am currently using Matlab's cscvn and smooth functions from the Curve Fitting Toolbox to try to achieve my goal, however I've found that for certain frames my code seems to work like I want it to (i.e fitting a curve to the object along its path), while for other frames the fitted curve does not match the path of the image. I've attached some examples comparing the processed image and the fitted curves (the blue circles represents the coordinates of the white pixels in the image, while the orange line represents the fitted curve).

Below is the accompanying code I created to make these curves, and I've also attached a link to the full code. One potential reason I've found for the poor fit has to do with the indexing of the white pixels which are later fed into the smooth and cscvn functions, however I don't know how I would address this. Am I approaching this problem the correct way, or should I be using different functions to try to create these curves? I feel like I'm starting to approach a wall with this, but at the same time it feels like I'm very close to achieving my goal with this problem.

%Reading image with some extra cropping%
    %I = sprintf('%i.jpeg',i);
    I = sprintf('3500.jpeg');
    img = imread(I);
    img = imbinarize(img);
    img(764:end,668:end) = 0;
    img = img(:,232:end);
    %imshow(img)
    %figure
    
%Indexing every white pixel from the image%
    [row,col] = size(img);
    xcoor = [];
    ycoor = [];
    for j = 1:col
        for k = 1:row
            if img(k,j) == 1
                xcoor = [xcoor, j];
                ycoor = [ycoor, -1*k];
            end
        end
    end
    
    %Smoothing Data and Curve Fitting%
    [xcoor, ii] = sort(xcoor);
    ycoor = ycoor(ii);
    yout = smooth(xcoor,ycoor,.1,'rloess');
    curve = cscvn([xcoor;yout']);
    subplot(2,1,1)
    scatter(xcoor,ycoor)
    hold on
    fnplt(curve);

[Abhay Iyer]

As a clarification, I'd like to fit these curves iteratively as my code iterates through a series of frames from a video of the object uncurling.

[Abhay Iyer]

Here are downloads to the above binary images if needed.

Answer 1

[Abhay Iyer]

        %Reading image%
        I = sprintf('%i.jpg',i);
        img = imread(I);
        img = imbinarize(img);

        %Get Endpoints%

        bwskel = bwmorph(img,'skel');
        bwend = bwmorph(bwskel,'endpoints');
        B = bwboundaries(bwend);
        b = numel(B);
        ep = [];
        for i = 1:b
            a = B{i,1};
            point = [a(1,2);a(1,1)];
            ep = [ep, point]; %Get endpoints from bwend%
        end
        last = ep(:,1);
        start = ep(:,2);

        %Index Points Along the Curve%

        r = start(2);
        c = start(1);
        contour = bwtraceboundary(img,[r c],'N',8); %Obtain coordinates of white pixels%
        [ii,jj] = unique(contour,'rows'); %Obtain indices of unique coord. to remove repeats%
        new = contour(sort(jj),:); %Removes repeats%
        x = new(:,2); 
        y = new(:,1);
        windowWidth = 169;
        polynomialOrder = 2;
        smoothX = sgolayfilt(x, polynomialOrder, windowWidth); %Smooth points using Savitzky-Golay filter for curve fitting%
        smoothY = sgolayfilt(y, polynomialOrder, windowWidth); %Smooth points using Savitzky-Golay filter for curve fitting%
        curve = [smoothX,smoothY];

The above code achieves my goal. Previously I used a double for loop to index the white pixels from my binary images, however this presented the problem of having the points being in a seemingly random sequence rather than following the path of the curve, leading to inconsistent curve fitting. Instead, I utilized MATLAB's bwboundaries, bwmorph, and bwtraceboundary functions to index the white pixels along the path of the curve, mainly referring to this previous answer. After I successfully indexed my pixels, I applied a Savitzky-Golay filter to smooth my points to create a smooth curve, which I'll later fit an equation to for curvature calculations using something like cscvn or John D'Errico's interparc function from the File Exchange.

Answer 2

[John D'Errico]

The curve fitting toolbox assumes the functions are SINGLE VALUED. There can only be ONE value of y for any x. If that is not true, for example, if you have replicates, then this is presumed to be noise. Therefore, you cannot directly use the curve fitting toolbox to model a function that is not single valued. Relationships (I won't call them functions, unless you are willing to call them implicit functions) that have an infinite slope are also not viable for the curve fitting toolbox. A curve that wraps over as you show is such an implicit function.

[Abhay Iyer]

Ok, that helps explain some of my issues, however if that is the case where should I go from from here? I'm not entirely sure what to search. Thanks for the response!

[John D'Errico]

Your problem is all you have are scattered points. It looks like you have captured them from an image, but they are not connected in any good sequence. The problem cscvn has, it assumes you know that sequence. But when you extract them as pixels from an image, they come out in a more arbitrary sequence, based on how find locates those points. That will surely fail on that curve that wraps curves around, and it will work on other simpler cases.

cscvn could handle the problem in theory, IF you just knew how to sequence those points along the curve itself.

I might decide to try an algorithm like CRUST, which can take those scattered points, and convert them into something that would now be managable, because it will essentially sequence the points along that curve. Once you have that, now cscvn will be happy. I think there is a CRUST implementation to be found on the file exchange, perhaps via GITHUB.

[Abhay Iyer]

Ok, I'll look into CRUST and provide updates, thank you for the suggestion. In general, is formatting the data in a way so cscvn is "happy" the best way to approach this type of problem, or is there a different method that makes more sense/is more applicable to trying to fit this relationship?

[Abhay Iyer]

Update: After searching for CRUST algorithm implementations I was unable to find something that could work well with my data, however I may just be misunderstanding how the algorithm actually works. Additionally, even after potentially formatting my data in such a way for the cscvn function to work properly, I am still lost on how I could smooth/filter this output to remove any potential noise/outliers in my pixel coordinates to create as smooth of a curve as possible for curvature calculations. Are there any further suggestions for how I could approach this problem?

[Abhay Iyer]

bump

[Wesley]

Hello, have you solved this problem? I also encountered a similar problem and would like to discuss it with you.